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Failure Of Brown Representability In Derived Categories
"... Let T be a triangulated category with coproducts, T c T the full subcategory of compact objects in T. If T is the homotopy category of spectra, Adams proved the following in [1]: All homological functors fT c g op ! Ab are the restrictions of representable functors on T, and all natural tr ..."
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Let T be a triangulated category with coproducts, T c T the full subcategory of compact objects in T. If T is the homotopy category of spectra, Adams proved the following in [1]: All homological functors fT c g op ! Ab are the restrictions of representable functors on T, and all natural transformations are the restrictions of morphisms in T. It has been something of a mystery, to what extent this generalises to other triangulated categories. In [36], it was proved that Adams' theorem remains true as long as T c is countable, but can fail in general. The failure exhibited was that there can be natural transformations not arising from maps in T. A puzzling open problem remained: Is every homological functor the restriction of a representable functor on T? In a recent paper, Beligiannis [5] made some progress. But in this article, we settle the problem. The answer is no. There are examples of derived categories T = D(R) of rings, and homological functors fT c g op ! Ab which are not restrictions of representables. Contents